Since \(RR^{\mathsf {T}} = I_{d} - e_{1}e_{1}^{\mathsf {T}}\) , we have
\(\lambda _{1}(\operatorname{\mathbb {E}}RAR^{\mathsf {T}}) = {I}(d^{-1} - (d^{3/2} + d)^{-1})\) .
Applying the matrix Chernoff inequality [1]}
we obtain
\(\Pr \left(\lambda _{2}(A) \ge 2{I}(d^{-1} - (d^{3/2} + d)^{-1})\right)&\le \Pr \left(\lambda _{1}(RAR^{\mathsf {T}}) \ge 2{I}(d^{-1} - (d^{3/2} + d)^{-1})\right)\\&\le d(e/4)^{{I}(d^{-1} + (d^{3/2} + d)^{-1})}.\) 
